Question

Let V be the set of real-valued functions that are defined at each x in the interval (−[infinity],[infinity]). If f​=f(x) and g​=g(x) are two functions in V and k is any scalar, we define the operations of addition and scalar multiplication by (f​+g​)(x)(kf​)(x)​=f(x)+g(x),=kf(x).​ Verify the Vector Space Axioms for the given set of vectors.

Answer 1

To verify the Vector Space Axioms for the given set of vectors, we need to show that the set of real-valued functions defined on the interval (-∞, ∞) under the operations of addition and scalar multiplication satisfies the vector space properties.

Step-by-step explanation:

To verify the Vector Space Axioms for the given set of vectors, we need to show that the set of real-valued functions defined on the interval (-∞, ∞) under the operations of addition and scalar multiplication satisfies the vector space properties.

1. Closure under Addition: The sum of two functions in V is also a real-valued function defined on (-∞, ∞). This can be shown by adding the functions f(x) and g(x) as (f+g)(x) = f(x) + g(x), which is still a real-valued function.
2. Commutativity of Addition: For any two functions f(x) and g(x) in V, (f+g)(x) = f(x) + g(x) = g(x) + f(x). The order of addition does not matter.
3. Associativity of Addition: For any three functions f(x), g(x), and h(x) in V, ((f+g)+h)(x) = (f+g)(x) + h(x) = f(x) + g(x) + h(x) is equal to (f+(g+h))(x) = f(x) + (g+h)(x) = f(x) + g(x) + h(x). The grouping of addition does not matter.
4. Existence of Zero Vector: There exists a function 0(x) = 0, which is the zero vector. For any function f(x) in V, (f+0)(x) = f(x) + 0 = f(x).
5. Existence of Additive Inverse: For any function f(x) in V, there exists a function -f(x) such that (f+(-f))(x) = f(x) + (-f)(x) = 0(x).
6. Closure under Scalar Multiplication: For any function f(x) in V and any scalar k, the product kf(x) is still a real-valued function defined on (-∞, ∞).
7. Compatibility of Scalar Multiplication with Field Multiplication: For any scalars k, l and any function f(x) in V, (kl)f(x) = k(lf(x)).
8. Identity Element of Scalar Multiplication: The function 1(x) = 1 is the identity element for scalar multiplication. For any function f(x) in V, (1f)(x) = 1*f(x) = f(x).

Since the set of real-valued functions defined on the interval (-∞, ∞) under the operations of addition and scalar multiplication satisfies all the vector space properties, it can be considered a vector space.